Quantum Error Correction Shyam Sundhar R Department of EE Mid Term Presentation, CS 682 Mid Term Presentation, CS 682 1 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Outline Shor Code 1 Decoherence Tackling Decoherence Error Correction by mapping into 2D spaces 2 Density Operator The theory Interaction 3 Interaction Operator Recovery Fundamentals of error correcting codes 4 Basic Definitions Recovery Operator State Independent Error Modeling Mid Term Presentation, CS 682 2 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Outline Shor Code 1 Decoherence Tackling Decoherence Error Correction by mapping into 2D spaces 2 Density Operator The theory Interaction 3 Interaction Operator Recovery Fundamentals of error correcting codes 4 Basic Definitions Recovery Operator State Independent Error Modeling Mid Term Presentation, CS 682 3 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
What is decoherence? How do we model it? Change in phase relation of different states due to environmental interaction. Mid Term Presentation, CS 682 4 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
What is decoherence? How do we model it? Change in phase relation of different states due to environmental interaction. | e 0 �| 0 � = ( | a 0 �| 0 � + | a 1 | 1 � ) | e 0 �| 1 � = ( | a 2 �| 0 � + | a 3 | 1 � ) where | e 0 � is the initial state of the environment and | a 0 � , | a 1 � , | a 2 � , | a 3 � are the final states of the environment. Mid Term Presentation, CS 682 4 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Outline Shor Code 1 Decoherence Tackling Decoherence Error Correction by mapping into 2D spaces 2 Density Operator The theory Interaction 3 Interaction Operator Recovery Fundamentals of error correcting codes 4 Basic Definitions Recovery Operator State Independent Error Modeling Mid Term Presentation, CS 682 5 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
encoding of qubits 1 | 0 S � = 2 ( | 000 � + | 111 � ) ⊗ ( | 000 � + | 111 � ) ⊗ ( | 000 � + | 111 � √ 2 1 | 1 S � = √ 2 ( | 000 � − | 111 � ⊗ ( | 000 � − | 111 � ) ⊗ ( | 000 � − | 111 � 2 Mid Term Presentation, CS 682 6 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
encoding of qubits 1 | 0 S � = 2 ( | 000 � + | 111 � ) ⊗ ( | 000 � + | 111 � ) ⊗ ( | 000 � + | 111 � √ 2 1 | 1 S � = √ 2 ( | 000 � − | 111 � ⊗ ( | 000 � − | 111 � ) ⊗ ( | 000 � − | 111 � 2 Decoherence of the first qubit √ ( | 0 S �− > 1 / 2)[( | a 0 �| 0 � + | a 1 | 1 � ) | 00 � + ( | a 2 �| 0 � + | a 3 | 1 � ) | 11 � )] √ ( | 1 S �− > 1 / 2)[( | a 0 �| 0 � + | a 1 | 1 � ) | 00 � − ( | a 2 �| 0 � + | a 3 | 1 � ) | 11 � )] Mid Term Presentation, CS 682 6 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
encoding of qubits √ 1 / 2 2( | a 0 � + | a 3 � )( | 000 � + | 111 � )+ √ 1 / 2 2( | a 0 � − | a 3 � )( | 000 � − | 111 � )+ √ 1 / 2 2( | a 1 � + | a 2 � )( | 100 � + | 011 � )+ √ 1 / 2 2( | a 1 � − | a 2 � )( | 100 � − | 011 � ) Mid Term Presentation, CS 682 7 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
encoding of qubits √ 1 / 2 2( | a 0 � + | a 3 � )( | 000 � + | 111 � )+ √ 1 / 2 2( | a 0 � − | a 3 � )( | 000 � − | 111 � )+ √ 1 / 2 2( | a 1 � + | a 2 � )( | 100 � + | 011 � )+ √ 1 / 2 2( | a 1 � − | a 2 � )( | 100 � − | 011 � ) √ 1 / 2 2( | a 0 � + | a 3 � )( | 000 � − | 111 � )+ √ 1 / 2 2( | a 0 � − | a 3 � )( | 000 � + | 111 � )+ √ 1 / 2 2( | a 1 � + | a 2 � )( | 100 � − | 011 � )+ √ 1 / 2 2( | a 1 � − | a 2 � )( | 100 � + | 011 � ) Mid Term Presentation, CS 682 7 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
′ b ′ c ′ � ). Projection of states in bell basis states( | abc � + | a Use of ancilla qubits to differentiate each subspace. Measurement of ancilla qubit. Application of appropriate operator. Takeaway The important thing to note is that the state of the environment is the same for corresponding vectors from the decoherence of the two quantum states encoding 0 and encoding 1. Since the corresponding vectors are same If it is projected to the superposition of the corresponding bell basis states appropriately the final state after correction tend to be in superposition For instance one projection would be to √ √ 1 / 2 2( | a 0 � − | a 3 � )( | 000 � − | 111 � ) + 1 / 2 2( | a 0 � − | a 3 � )( | 000 � + | 111 � ) Mid Term Presentation, CS 682 8 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Outline Shor Code 1 Decoherence Tackling Decoherence Error Correction by mapping into 2D spaces 2 Density Operator The theory Interaction 3 Interaction Operator Recovery Fundamentals of error correcting codes 4 Basic Definitions Recovery Operator State Independent Error Modeling Mid Term Presentation, CS 682 9 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
What is density operator? Density operators Positive semidefinite operators having trace equal to 1 are called density operators. D ( X ) = ρ ∈ Pos ( X ) : Tr ( ρ ) = 1 Mid Term Presentation, CS 682 10 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
What is density operator? Density operators Positive semidefinite operators having trace equal to 1 are called density operators. D ( X ) = ρ ∈ Pos ( X ) : Tr ( ρ ) = 1 D(X) will be used to denote the collection of density operators acting on a complex Euclidean space X . Mid Term Presentation, CS 682 10 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
What is density operator? Density operators Positive semidefinite operators having trace equal to 1 are called density operators. D ( X ) = ρ ∈ Pos ( X ) : Tr ( ρ ) = 1 D(X) will be used to denote the collection of density operators acting on a complex Euclidean space X . For any complex Euclidean space X , the set D(X ) of density operators acting on X is convex and compact. The extreme points of D(X ) coincide with the rank-one projection operators. These are the operators of the form uu ∗ for u ∈ S(X ) being a unit vector. Mid Term Presentation, CS 682 10 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Quantum States as Density operators Definition A quantum state is a density operator of the form ρ ∈ D(X ) for some choice of a complex Euclidean space X Mid Term Presentation, CS 682 11 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Quantum States as Density operators Definition A quantum state is a density operator of the form ρ ∈ D(X ) for some choice of a complex Euclidean space X Pure state A quantum state ρ ∈ D ( X ) is said to be a pure state if it has rank equal to 1. Equivalently, ρ is a pure state if there exists a unit vector u ∈ X such that ρ = uu ∗ Mid Term Presentation, CS 682 11 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Quantum States as Density operators Definition A quantum state is a density operator of the form ρ ∈ D(X ) for some choice of a complex Euclidean space X Pure state A quantum state ρ ∈ D ( X ) is said to be a pure state if it has rank equal to 1. Equivalently, ρ is a pure state if there exists a unit vector u ∈ X such that ρ = uu ∗ Theorem state representation ρ = Σ p ( a ) ρ a Mid Term Presentation, CS 682 11 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Quantum States as Density operators Definition A quantum state is a density operator of the form ρ ∈ D(X ) for some choice of a complex Euclidean space X Pure state A quantum state ρ ∈ D ( X ) is said to be a pure state if it has rank equal to 1. Equivalently, ρ is a pure state if there exists a unit vector u ∈ X such that ρ = uu ∗ Theorem state representation ρ = Σ p ( a ) ρ a Unitary Operation Any state ρ f after Unitary Operation A a can be written as ρ f = A a ρ i A ∗ a Mid Term Presentation, CS 682 11 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
Outline Shor Code 1 Decoherence Tackling Decoherence Error Correction by mapping into 2D spaces 2 Density Operator The theory Interaction 3 Interaction Operator Recovery Fundamentals of error correcting codes 4 Basic Definitions Recovery Operator State Independent Error Modeling Mid Term Presentation, CS 682 12 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
the main idea mapping | ψ � into a higher dimensional Hilbert space (using ancilla qubits which are assumed to be in their | 0 � states initially): ( α | 0 � + β | 1 �| 000 ... � α | 0 L � + β | 1 L � . error induced maps the new state into one of a family of two-dimensional subspaces which preserve the relative coherence of the quantum information (i.e. in each subspace, the state of the computer should be in a tensor product state with the environment). A measurement is then performed which projects the state into one of these subspaces Mid Term Presentation, CS 682 13 / Shyam Sundhar R (IIT K) Quantum Error Correction 32
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